SECTION 10.1 665 - NCKU

SECTION 10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS 665

1–4 Sketch the curve by using the parametric equations to plotpoints. Indicate with an arrow the direction in which the curve istraced as increases.

1. , ,

2. , ,

3. , ,

4. , ,

5–10(a) Sketch the curve by using the parametric equations to plot

points. Indicate with an arrow the direction in which the curveis traced as t increases.

(b) Eliminate the parameter to find a Cartesian equation of thecurve.

5. ,

6. , ,

7. , ,

8. , ,

t

x � cos2t y � 1 � sin t 0 � t � ��2

x � e�t � t y � e t � t �2 � t � 2

x � 3 � 4t y � 2 � 3t

x � 1 � 2t y � 12t � 1 �2 � t � 4

x � 1 � t 2 y � t � 2 �2 � t � 2

�2 � t � 2y � t 3 � 1x � t � 1

x � 1 � st y � t 2 � 4 t 0 � t � 5

x � 2 cos t y � t � cos t 0 � t � 2�

9. ,

10. ,

11–18(a) Eliminate the parameter to find a Cartesian equation of the

curve.(b) Sketch the curve and indicate with an arrow the direction in

which the curve is traced as the parameter increases.

11. , ,

12. , ,

13. , ,

14. ,

15. ,

16. ,

17. ,

18. , ,

x � t 2 y � t 3

x � st y � 1 � t

x � sin 12� y � cos 12� ��

x � 12 cos � y � 2 sin � 0 � � � �

x � sin t y � csc t 0 � t � ��2

x � et � 1 y � e 2 t

x � e 2 t y � t � 1

y � st � 1y � st � 1

x � sinh t y � cosh t

x � tan2� y � sec � ��2 � � � ��2

10.1 Exercises

; Graphing calculator or computer required 1. Homework Hints available at stewartcalculus.com

When , both branches are smooth; but when reaches , the right branchacquires a sharp point, called a cusp. For between and 0 the cusp turns into a loop,which becomes larger as approaches 0. When , both branches come together andform a circle (see Example 2). For between 0 and 1, the left branch has a loop, whichshrinks to become a cusp when . For , the branches become smooth again,and as increases further, they become less curved. Notice that the curves with posi-tive are reflections about the -axis of the corresponding curves with negative.

These curves are called conchoids of Nicomedes after the ancient Greek scholarNicomedes. He called them conchoids because the shape of their outer branches resembles that of a conch shell or mussel shell.

a=_2 a=_1 a=_0.5 a=_0.2

a=2a=1a=0.5a=0

a � �1 a �1a �1

a a � 0a

a � 1 a � 1a a

y a

FIGURE 17 Members of the familyx=a+cos t, y=a tan t+sin t,all graphed in the viewing rectangle�_4, 4� by �_4, 4�

98845_ch10_ptg01_hr_659-669.qk_98845_ch10_ptg01_hr_659-669 8/18/11 1:53 PM Page 665

user

螢光標示

user

螢光標示

user

螢光標示

user

螢光標示

user

螢光標示

666 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES

19–22 Describe the motion of a particle with position as varies in the given interval.

19. , ,

20. , ,

21. , ,

22. , ,

23. Suppose a curve is given by the parametric equations ,, where the range of is and the range of is

. What can you say about the curve?

24. Match the graphs of the parametric equations andin (a)–(d) with the parametric curves labeled I–IV.

Give reasons for your choices.

�x, y�t

x � 3 � 2 cos t y � 1 � 2 sin t ��2 � t � 3��2

x � 2 sin t y � 4 � cos t 0 � t � 3��2

x � 5 sin t y � 2 cos t �� t � 5�

x � sin t y � cos2t �2� � t � 2�

x � f �t�y � t�t� f �1, 4� t

�2, 3�

x � f �t�y � t�t�

t

x

2

1

1

t

y

1

1

y

x

2

2

(a) I

(b) IIx

t

2

1 t

2

1

y y

x

2

2

(c) III

t

2

2

yx

t

2

2

(d) IV

t

2

2

yx

t

2

2

y

x

2

2

1

y

x

1

2

25–27 Use the graphs of and to sketch the para-metric curve , . Indicate with arrows the directionin which the curve is traced as increases.

25.

26.

27.

28. Match the parametric equations with the graphs labeled I-VI.Give reasons for your choices. (Do not use a graphing device.)(a) ,

(b) ,

(c) ,

(d) ,

(e) ,

(f ) ,

y � t�t�x � f �t�y � t�t�x � f �t�

t

t

x

_1

1 t

y

1

1

t

x

1

1 t

y

1

1

t

y

1

1t

x

1

1

y � t 2x � t 4 � t � 1

y � stx � t 2 � 2t

y � sin�t � sin 2t�x � sin 2t

y � sin 2tx � cos 5t

y � t 2 � cos 3tx � t � sin 4t

y �cos 2t

4 � t 2x �sin 2t

4 � t 2

x

y

x

y

x

y

x

y

x

y

x

y

I II III

IV V VI


user

螢光標示

SECTION 10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS 667

; 29. Graph the curve .

; 30. Graph the curves and and findtheir points of intersection correct to one decimal place.

31. (a) Show that the parametric equations

where , describe the line segment that joins thepoints and .

(b) Find parametric equations to represent the line segmentfrom to .

; 32. Use a graphing device and the result of Exercise 31(a) todraw the triangle with vertices , , and .

33. Find parametric equations for the path of a particle thatmoves along the circle in the mannerdescribed.(a) Once around clockwise, starting at (b) Three times around counterclockwise, starting at (c) Halfway around counterclockwise, starting at

; 34. (a) Find parametric equations for the ellipse. [Hint: Modify the equations of

the circle in Example 2.](b) Use these parametric equations to graph the ellipse when

and b � 1, 2, 4, and 8.(c) How does the shape of the ellipse change as b varies?

; 35–36 Use a graphing calculator or computer to reproduce thepicture.

35. 36.

37–38 Compare the curves represented by the parametricequations. How do they differ?

37. (a) , (b) , (c) ,

38. (a) , (b) , (c) ,

39. Derive Equations 1 for the case .

40. Let be a point at a distance from the center of a circle ofradius . The curve traced out by as the circle rolls along astraight line is called a trochoid. (Think of the motion of apoint on a spoke of a bicycle wheel.) The cycloid is the spe-cial case of a trochoid with . Using the same parameter

as for the cycloid and, assuming the line is the -axis and

x � y � 2 sin �y

x � y 3 � 4yy � x 3 � 4x

y � y1 � �y2 � y1�tx � x1 � �x 2 � x1�t

0 � t � 1P2�x 2, y2 �P1�x1, y1�

�3, �1��2, 7�

C �1, 5�B �4, 2�A �1, 1�

x 2 � �y � 1�2 � 4

�2, 1��2, 1�

�0, 3�

x 2�a 2 � y 2�b 2 � 1

a � 3

0

y

x

2

3 8

4

0

2

y

x2

y � t 4x � t 6y � t 2x � t 3

y � e�2 tx � e�3 t

y � sec2tx � cos ty � t �2x � ty � e�2 tx � e t

��2 � � � �

dPPr

d � rx�

when is at one of its lowest points, show that para-metric equations of the trochoid are

Sketch the trochoid for the cases and .

41. If and are fixed numbers, find parametric equations forthe curve that consists of all possible positions of the pointin the figure, using the angle as the parameter. Then elimi-nate the param eter and identify the curve.

42. If and are fixed numbers, find parametric equations forthe curve that consists of all possible positions of the pointin the figure, using the angle as the parameter. The linesegment is tangent to the larger circle.

43. A curve, called a witch of Maria Agnesi, consists of all pos-sible positions of the point in the figure. Show that para-metric equations for this curve can be written as

Sketch the curve.

P� � 0

y � r � d cos �x � r� � d sin �

d � rd � r

baP

�

O

y

x¨

ab P

baP

�AB

O x

y

¨

ab

A

B

P

P

y � 2a sin2�x � 2a cot �

O x

a

A P

y=2a

¨

yC


L A B O R AT O R Y P R O J E C T ; RUNNING CIRCLES AROUND CIRCLES

In this project we investigate families of curves, called hypocycloids and epicycloids, that aregenerated by the motion of a point on a circle that rolls inside or outside another circle.

1. A hypocycloid is a curve traced out by a fixed point P on a circle C of radius b as C rolls on theinside of a circle with center O and radius a. Show that if the initial position of P is andthe parameter is chosen as in the figure, then parametric equations of the hypocycloid are

�a, 0��

y � �a � b� sin � � b sin�a � b

b�x � �a � b� cos � � b cos�a � b

b�

; Graphing calculator or computer required


44. (a) Find parametric equations for the set of all points asshown in the figure such that . (This curveis called the cissoid of Diocles after the Greek scholarDiocles, who introduced the cissoid as a graphicalmethod for constructing the edge of a cube whose volumeis twice that of a given cube.)

(b) Use the geometric description of the curve to draw arough sketch of the curve by hand. Check your work byusing the parametric equations to graph the curve.

; 45. Suppose that the position of one particle at time is given by

and the position of a second particle is given by

(a) Graph the paths of both particles. How many points ofintersection are there?

(b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place atthe same time? If so, find the collision points.

(c) Describe what happens if the path of the second particleis given by

46. If a projectile is fired with an initial velocity of meters persecond at an angle above the horizontal and air resistanceis assumed to be negligible, then its position after seconds

P

� OP � � � AB �

xO

y

A

Px=2a

B

a

t

0 � t � 2�y1 � 2 cos tx1 � 3 sin t

0 � t � 2�y2 � 1 � sin tx 2 � �3 � cos t

x 2 � 3 � cos t y2 � 1 � sin t 0 � t � 2�

v0

t

is given by the parametric equations

where is the acceleration due to gravity ( m�s ).(a) If a gun is fired with and m�s, when

will the bullet hit the ground? How far from the gun willit hit the ground? What is the maximum height reached by the bullet?

; (b) Use a graphing device to check your answers to part (a).Then graph the path of the projectile for several other values of the angle to see where it hits the ground.Summarize your findings.

(c) Show that the path is parabolic by eliminating the parameter.

; 47. Investigate the family of curves defined by the parametricequations , . How does the shape change as increases? Illustrate by graphing several members of thefamily.

; 48. The swallowtail catastrophe curves are defined by the para-metric equations , . Graphseveral of these curves. What features do the curves have incommon? How do they change when increases?

; 49. Graph several members of the family of curves withparametric equations , , where

. How does the shape change as increases? For whatvalues of does the curve have a loop?

; 50. Graph several members of the family of curves, where is a positive

integer. What features do the curves have in common? Whathappens as increases?

; 51. The curves with equations , arecalled Lissajous figures. Investigate how these curves varywhen , , and vary. (Take to be a positive integer.)

; 52. Investigate the family of curves defined by the parametricequations , , where . Start by letting be a positive integer and see what happens to theshape as increases. Then explore some of the possibilitiesthat occur when is a fraction.

� 30 v0 � 500

x � t 2 y � t 3 � ctc

x � 2ct � 4t 3 y � �ct 2 � 3t 4

c

x � t � a cos t y � t � a sin ta � 0 a

a

x � sin t � sin nt ny � cos t � cos nt

n

x � a sin nt y � b cos t

a b n n

x � cos t y � sin t � sin ct c � 0c

cc

29.8t

y � �v0 sin �t �12 tt 2x � �v0 cos �t

xO

y

a

C

Pb(a, 0)¨

A


SECTION 10.2 CALCULUS WITH PARAMETRIC CURVES 675

1–2 Find .

1. , 2. ,

3–6 Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

3. , ;

4. , ;

5. , ;

6. , ;

7–8 Find an equation of the tangent to the curve at the givenpoint by two methods: (a) without eliminating the parameter and(b) by first eliminating the parameter.

7. , ;

8. , ;

; 9–10 Find an equation of the tangent(s) to the curve at the givenpoint. Then graph the curve and the tangent(s).

9. , ;

10. , ;

11–16 Find and . For which values of is thecurve concave upward?

11. , 12. ,

13. , 14. ,

15. , ,

16. , ,

17–20 Find the points on the curve where the tangent is horizon-tal or vertical. If you have a graphing device, graph the curve tocheck your work.

17. ,

18. ,

19. ,

20. ,

; 21. Use a graph to estimate the coordinates of the rightmost pointon the curve , . Then use calculus to find theexact coordinates.

; 22. Use a graph to estimate the coordinates of the lowest pointand the leftmost point on the curve , .Then find the exact coordinates.

dy�dx

x � t sin t y � t 2 � t x � 1�t y � st e�t

x � t � t�1 y � 1 � t 2 t � 1

x � t cos t y � t sin t t � �

x � 1 � ln t y � t 2 � 2 �1, 3�

x � 1 � st y � et2

�2, e�

x � 6 sin t y � t 2 � t �0, 0�

x � cos t � cos 2t y � sin t � sin 2t ��1, 1�

dy�dx d 2 y�dx 2 t

x � t 2 � 1 y � t 2 � t

x � e t y � te� t x � t 2 � 1 y � e t � 1

x � 2 sin t y � 3 cos t 0 � t � 2�

x � cos 2 t y � cos t 0 � t � �

x � t 3 � 3t y � t 2 � 3

x � t 3 � 3t y � t 3 � 3t 2

x � cos � y � cos 3�

x � e sin � y � e cos �

x � t � t 6 y � e t

x � t 4 � 2t y � t � t 4

x � t 4 � 1 y � t 3 � t t � �1

x � cos � � sin 2� y � sin � � cos 2� � � 0

x � t 3 � 12t y � t 2 � 1

; 23–24 Graph the curve in a viewing rectangle that displays allthe important aspects of the curve.

23. ,

24. ,

25. Show that the curve , has twotangents at and find their equations. Sketch the curve.

; 26. Graph the curve , todiscover where it crosses itself. Then find equations of bothtangents at that point.

27. (a) Find the slope of the tangent line to the trochoid, in terms of . (See

Exercise 40 in Section 10.1.)(b) Show that if , then the trochoid does not have a

vertical tangent.

28. (a) Find the slope of the tangent to the astroid ,in terms of . (Astroids are explored in the

Laboratory Project on page 668.)(b) At what points is the tangent horizontal or vertical?(c) At what points does the tangent have slope 1 or ?

29. At what points on the curve , doesthe tangent line have slope ?

30. Find equations of the tangents to the curve ,that pass through the point .

31. Use the parametric equations of an ellipse, ,, , to find the area that it encloses.

32. Find the area enclosed by the curve , andthe .

33. Find the area enclosed by the and the curve , .

34. Find the area of the region enclosed by the astroid, . (Astroids are explored in the Labo-

ratory Project on page 668.)

35. Find the area under one arch of the trochoid of Exercise 40 inSection 10.1 for the case .

x � t 4 � 2t 3 � 2t 2 y � t 3 � t

x � t 4 � 4t 3 � 8t 2 y � 2t 2 � t

x � cos t y � sin t cos t�0, 0�

x � cos t � 2 cos 2t y � sin t � 2 sin 2t

x � r� � d sin � y � r � d cos � �

d � r

x � a cos3�y � a sin3� �

�1

x � 2t 3 y � 1 � 4t � t 2

1

x � 3t 2 � 1y � 2t 3 � 1 �4, 3�

x � a cos �y � b sin � 0 � � � 2�

x � t 2 � 2t y � sty-axis

x-axisx � 1 � e t y � t � t 2

x � a cos3� y � a sin3�

y

x0 a_a

_a

a

d � r

10.2 Exercises

; Graphing calculator or computer required Computer algebra system required 1. Homework Hints available at stewartcalculus.comCAS


user

螢光標示

user

螢光標示

user

螢光標示

user

螢光標示

user

螢光標示


36. Let be the region enclosed by the loop of the curve inExample 1.(a) Find the area of .(b) If is rotated about the -axis, find the volume of the

resulting solid.(c) Find the centroid of .

37–40 Set up an integral that represents the length of the curve.Then use your calculator to find the length correct to fourdecimal places.

37. , ,

38. , ,

39. , ,

40. , ,

41–44 Find the exact length of the curve.

41. , ,

42. , ,

43. , ,

44. , ,

; 45–46 Graph the curve and find its length.

45. , ,

46. , ,

; 47. Graph the curve , and find itslength correct to four decimal places.

48. Find the length of the loop of the curve ,.

49. Use Simpson’s Rule with to estimate the length of thecurve , , .

50. In Exercise 43 in Section 10.1 you were asked to derive theparametric equations , for thecurve called the witch of Maria Agnesi. Use Simpson’s Rulewith to estimate the length of the arc of this curvegiven by .

51–52 Find the distance traveled by a particle with positionas varies in the given time interval. Compare with the length ofthe curve.

51. , ,

52. , ,

53. Show that the total length of the ellipse ,, , is

�� x

�

x � t � e�t y � t � e�t 0 � t � 2

x � t 2 � t y � t 4 1 � t � 4

x � t � 2 sin t y � 1 � 2 cos t 0 � t � 4�

x � t � st y � t � st 0 � t � 1

�

0 � t � 1y � 4 � 2t 3x � 1 � 3t 2

0 � t � 3y � 5 � 2tx � et � e�t

0 � t � 1y � t cos tx � t sin t

0 � t � �y � 3 sin t � sin 3tx � 3 cos t � cos 3t

0 � t � �y � e t sin tx � e t cos t

��4 � t � 3��4y � sin tx � cos t � ln(tan 12 t)

x � sin t � sin 1.5t y � cos t

x � 3t � t 3

y � 3t 2

n � 6�6 � t � 6y � t � e tx � t � e t

y � 2a sin2�x � 2a cot �

n � 4��4 � � � ��2

�x, y�t

0 � t � 3�y � cos2tx � sin2t

0 � t � 4�y � cos tx � cos2t

x � a sin �a � b � 0y � b cos �

L � 4a y��2

0s1 � e 2 sin2� d�

where is the eccentricity of the ellipse , where.

54. Find the total length of the astroid , ,where

55. (a) Graph the epitrochoid with equations

What parameter interval gives the complete curve?(b) Use your CAS to find the approximate length of this

curve.

56. A curve called Cornu’s spiral is defined by the parametricequations

where and are the Fresnel functions that were intro ducedin Chapter 4.(a) Graph this curve. What happens as and as

?(b) Find the length of Cornu’s spiral from the origin to the

point with parameter value .

57–60 Set up an integral that represents the area of the surfaceobtained by rotating the given curve about the -axis. Then useyour calculator to find the surface area correct to four decimalplaces.

57. , ,

58. , ,

59. , ,

60. , ,

61–63 Find the exact area of the surface obtained by rotating thegiven curve about the -axis.

61. , ,

62. , ,

63. , ,

; 64. Graph the curve

If this curve is rotated about the -axis, find the area of theresulting surface. (Use your graph to help find the correct parameter interval.)

65–66 Find the surface area generated by rotating the given curveabout the -axis.

65. , ,

(e � c�aec � sa 2 � b 2 )

y � a sin3�x � a cos3�a � 0.

CAS

x � 11 cos t � 4 cos�11t�2�

y � 11 sin t � 4 sin�11t�2�

CAS

x � C�t� � yt

0cos��u 2�2� du

y � S�t� � yt

0sin��u 2�2� du

SC

t l t l �

t

x

0 � t � ��2y � t cos tx � t sin t

0 � t � ��2y � sin 2tx � sin t

0 � t � 1y � �t 2 � 1�e tx � 1 � te t

0 � t � 1y � t � t 4x � t 2 � t 3

x

0 � t � 1y � t 2x � t 3

0 � t � 1y � 3t 2x � 3t � t 3

0 � � � ��2y � a sin3�x � a cos3�

y � 2 sin � � sin 2�x � 2 cos � � cos 2�

x

y

0 � t � 5y � 2t 3x � 3t 2


LABORATORY PROJECT BÉZIER CURVES 677

66. , ,

67. If is continuous and for , show that theparametric curve , , , can be put inthe form . [Hint: Show that exists.]

68. Use Formula 2 to derive Formula 7 from Formula 8.2.5 for thecase in which the curve can be represented in the form

, .

69. The curvature at a point of a curve is defined as

where is the angle of inclination of the tangent line at , as shown in the figure. Thus the curvature is the absolute valueof the rate of change of with respect to arc length. It can beregarded as a measure of the rate of change of direction of thecurve at and will be studied in greater detail in Chapter 13.(a) For a parametric curve , , derive the

formula

where the dots indicate derivatives with respect to , so. [Hint: Use and Formula 2 to

find . Then use the Chain Rule to find .](b) By regarding a curve as the parametric curve

, , with parameter , show that the formulain part (a) becomes

0 � t � 1y � 4e t�2x � e t � t

a � t � bf ��t� � 0f �a � t � by � t�t�x � f �t�f �1y � F�x�

a � x � by � F�x�

P

� � � d�

ds �P�

�

Py � y�t�x � x�t�

� � x�y�� x��y� x� 2 � y� 2 �3�2

t� � tan�1�dy�dx�x� � dx�dt

d��dsd��dty � f �x�

xy � f �x�x � x

� � d 2 y�dx 2 1 � �dy�dx�2 �3�2

0 x

y

P

˙

70. (a) Use the formula in Exercise 69(b) to find the curvature ofthe parabola at the point .

(b) At what point does this parabola have maximum curvature?

71. Use the formula in Exercise 69(a) to find the curvature of thecycloid , at the top of one of itsarches.

72. (a) Show that the curvature at each point of a straight line is .

(b) Show that the curvature at each point of a circle of radius is .

73. A string is wound around a circle and then unwound whilebeing held taut. The curve traced by the point at the end ofthe string is called the involute of the circle. If the circle hasradius and center and the initial position of is , andif the parameter is chosen as in the figure, show thatparametric equations of the involute are

74. A cow is tied to a silo with radius by a rope just long enoughto reach the opposite side of the silo. Find the area available forgrazing by the cow.

r O P �r, 0��

x � r �cos � � � sin �� y � r �sin � � � cos ��

xO

y

r

¨ P

T

r

�1, 1�y � x 2

y � 1 � cos �x � � � sin �

� � 0

� � 1�rr

P

L A B O R AT O R Y P R O J E C T ; BÉZIER CURVES

Bézier curves are used in computer-aided design and are named after the French mathema-tician Pierre Bézier (1910–1999), who worked in the automotive industry. A cubic Bézier curve is determined by four control points, and , and is defined by the parametric equations

P0�x0, y0 �, P1�x1, y1�, P2�x2, y2 �, P3�x3, y3 �

x � x0�1 � t�3 � 3x1t�1 � t�2 � 3x2t 2�1 � t� � x3t 3

y � y0�1 � t�3 � 3y1t�1 � t�2 � 3y2t 2�1 � t� � y3t 3





1–2 Plot the point whose polar coordinates are given. Then findtwo other pairs of polar coordinates of this point, one withand one with .

1. (a) (b) (c)

2. (a) (b) (c)

3–4 Plot the point whose polar coordinates are given. Then find theCartesian coordinates of the point.

3. (a) (b) (c)

r � 0r � 0

��1, ��2��1, �3��4��2, ��3�

�1, �1��3, ��6��1, 7��4�

��2, 3��4�(2, �2��3)�1, ��

4. (a) (b) (c)

5–6 The Cartesian coordinates of a point are given.(i) Find polar coordinates of the point, where and .(ii) Find polar coordinates of the point, where and .

5. (a) (b)

6. (a) (b)

�r, �� r � 00 � � � 2�

�r, �� r � 00 � � � 2�

�2, �2� (�1, s3 )(3s3 , 3) �1, �2�

�2, �7��6��1, 5��2�(�s2 , 5��4)

10.3 Exercises

Switching from to , we have the equations

and Figure 18 shows the resulting curve. Notice that this rose has 16 loops.

Investigate the family of polar curves given by . Howdoes the shape change as changes? (These curves are called limaçons, after a Frenchword for snail, because of the shape of the curves for certain values of .)

SOLUTION Figure 19 shows computer-drawn graphs for various values of . Forthere is a loop that decreases in size as decreases. When the loop disappears andthe curve becomes the cardioid that we sketched in Example 7. For between and thecardioid’s cusp is smoothed out and becomes a “dimple.” When de creases from to ,the limaçon is shaped like an oval. This oval becomes more circular as , and when

the curve is just the circle .

The remaining parts of Figure 19 show that as becomes negative, the shapes changein reverse order. In fact, these curves are reflections about the horizontal axis of the corre-sponding curves with positive .

Limaçons arise in the study of planetary motion. In particular, the trajectory of Mars, asviewed from the planet Earth, has been modeled by a limaçon with a loop, as in the partsof Figure 19 with .

x � sin�8t�5� cos t y � sin�8t�5� sin t 0 � t � 10�

r � 1 � c sin �c

c

c c � 1c c � 1

c 1 12

c 12 0

c l 0c � 0 r � 1

c

c

EXAMPLE 11v

t�

� c � � 1

1

_1

_1 1

FIGURE 18r=sin(8¨/5)

c=2.5

FIGURE 19Members of the family oflimaçons r=1+c sin ̈

c=0 c=_0.2 c=_0.5 c=_0.8 c=_1

c=_2

c=1.7 c=1 c=0.7 c=0.5 c=0.2

In Exercise 53 you are asked to prove analyticallywhat we have discovered from the graphs in Figure 19.


user

螢光標示

user

螢光標示

SECTION 10.3 POLAR COORDINATES 687

7–12 Sketch the region in the plane consisting of points whosepolar coordinates satisfy the given conditions.

7.

8. ,

9. ,

10. ,

11. ,

12. ,

13. Find the distance between the points with polar coordinatesand .

14. Find a formula for the distance between the points with polarcoordinates and .

15–20 Identify the curve by finding a Cartesian equation for thecurve.

15. 16.

17. 18.

19. 20.

21–26 Find a polar equation for the curve represented by the givenCartesian equation.

21. 22.

23. 24.

25. 26.

27–28 For each of the described curves, decide if the curve wouldbe more easily given by a polar equation or a Cartesian equation.Then write an equation for the curve.

27. (a) A line through the origin that makes an angle of withthe positive -axis

(b) A vertical line through the point

28. (a) A circle with radius 5 and center (b) A circle centered at the origin with radius 4

29–46 Sketch the curve with the given polar equation by firstsketching the graph of as a function of in Cartesian coordinates.

29. 30.

31. 32.

33. , 34. ,

35. 36.

37. 38.

39. 40.

0 � r � 2 � � � � 3��2

r 0 ��4 � � � 3��4

1 � r � 3 ��6 � � � 5��6

2 � r � 3 5��3 � � � 7��3

r 1 � � � � 2�

�4, 2��3��2, ��3�

�r2, �2 ��r1, �1�

� � ��3r � 2 cos �

r 2 cos 2� � 1 r � tan � sec �

y � xy � 2

4y 2 � xy � 1 � 3x

x 2 � y 2 � 2cx xy � 4

��6x

�3, 3�

�2, 3�

r � 1 � cos �r � �2 sin �

r �

r � 2�1 � cos ��

r � � � 0 r � ln � � 1

r � cos 5�r � 4 sin 3�

r � 3 cos 6�r � 2 cos 4�

r � 2 � sin �r � 1 � 2 sin �

r � 1 � 2 cos �

1 � r � 2

r � 2 r cos � � 1

41. 42.

43. 44.

45. 46.

47–48 The figure shows a graph of as a function of in Cartesiancoordinates. Use it to sketch the corresponding polar curve.

47. 48.

49. Show that the polar curve (called a conchoid)has the line as a vertical asymptote by showing that

. Use this fact to help sketch the conchoid.

50. Show that the curve (also a conchoid) has theline as a horizontal asymptote by showing that

. Use this fact to help sketch the conchoid.

51. Show that the curve (called a cissoid of Diocles) has the line as a vertical asymptote. Show alsothat the curve lies entirely within the vertical strip .Use these facts to help sketch the cissoid.

52. Sketch the curve .

53. (a) In Example 11 the graphs suggest that the limaçonhas an inner loop when . Prove

that this is true, and find the values of that correspond tothe inner loop.

(b) From Figure 19 it appears that the limaçon loses its dimplewhen . Prove this.

54. Match the polar equations with the graphs labeled I–VI. Givereasons for your choices. (Don’t use a graphing device.)

(a) (b)(c) (d)(e) (f )

r � 2 � sin 3� r 2� � 1

r � 1 � 2 cos 2� r � 3 � 4 cos �

r �

¨

r

0 π 2π

2

_2¨

r

0 π 2π

1

2

r � 4 � 2 sec �x � 2

lim r l� x � 2

r � 2 � csc �y � �1

lim r l� y � �1

r � sin � tan �x � 1

0 � x � 1

�x 2 � y 2 �3 � 4x 2 y 2

r � 1 � c sin � � c � � 1�

c � 12

r � s� , 0 � � � 16� r � � 2, 0 � � � 16�

r � cos��3� r � 1 � 2 cos �r � 2 � sin 3� r � 1 � 2 sin 3�

I II III

IV V VI

r 2 � cos 4�r 2 � 9 sin 2�


user

螢光標示

user

螢光標示

user

螢光標示

user

螢光標示

user

螢光標示

user

螢光標示

user

螢光標示


55–60 Find the slope of the tangent line to the given polar curveat the point specified by the value of .

55. , 56. ,

57. , 58. ,

59. , 60. ,

61–64 Find the points on the given curve where the tangent lineis horizontal or vertical.

61. 62.

63. 64.

65. Show that the polar equation , where, represents a circle, and find its center and radius.

66. Show that the curves and intersect atright angles.

; 67–72 Use a graphing device to graph the polar curve. Choosethe parameter interval to make sure that you produce the entirecurve.

67. (nephroid of Freeth)

68. (hippopede)

69. (butterfly curve)

70. (valentine curve)

71. (PacMan curve)

72.

; 73. How are the graphs of andrelated to the graph of ?

In general, how is the graph of related to thegraph of ?

�

� � ��3r � 2 � sin �� 6r � 2 sin �

r � 1�� r � cos��3� � � �

r � cos 2� � � ��4 r � 1 � 2 cos� � � ��3

r � 3 cos � r � 1 � sin �

r � 1 � cos � r � e �

r � a sin � � b cos �ab � 0

r � a cos �r � a sin �

r � 1 � 2 sin��2�

r � s1 � 0.8 sin 2�

r � e sin � � 2 cos�4��

r � � tan � �� cot � �

r � 1 � cos999�

r � sin2�4�� cos�4��

r � 1 � sin�� 6�r � 1 � sin �r � 1 � sin�� 3�

r � f �� r � f ��

; 74. Use a graph to estimate the -coordinate of the highest pointson the curve . Then use calculus to find the exactvalue.

; 75. Investigate the family of curves with polar equations, where is a real number. How does the

shape change as changes?

; 76. Investigate the family of polar curves

where is a positive integer. How does the shape change asincreases? What happens as becomes large? Explain theshape for large by considering the graph of as a functionof in Cartesian coordinates.

77. Let be any point (except the origin) on the curve .If is the angle between the tangent line at and the radialline , show that

[Hint: Observe that in the figure.]

78. (a) Use Exercise 77 to show that the angle between the tan-gent line and the radial line is at every point onthe curve .

; (b) Illustrate part (a) by graphing the curve and the tangentlines at the points where and .

(c) Prove that any polar curve with the property thatthe angle between the radial line and the tangent line isa constant must be of the form , where andare constants.

OP

tan �r

dr�d�

� � � �

O

P

ÿ

¨ ˙

r=f(¨ )

� ��4r � e�

� � 0 ��2r � f ��

r � Ce k� C k

yr � sin 2�

r � 1 � c cos � cc

r � 1 � cosn�

n nn

n r�

r � f ��PP

L A B O R AT O R Y P R O J E C T ; FAMILIES OF POLAR CURVES

In this project you will discover the interesting and beautiful shapes that members of families ofpolar curves can take. You will also see how the shape of the curve changes when you vary theconstants.

1. (a) Investigate the family of curves defined by the polar equations , where is apositive integer. How is the number of loops related to ?

(b) What happens if the equation in part (a) is replaced by ?

2. A family of curves is given by the equations , where is a real number and is a positive integer. How does the graph change as increases? How does it change as

changes? Illustrate by graphing enough members of the family to support your conclusions.

r � sin n� nnr � � sin n� �

r � 1 � c sin n� cn n c




so, using , we have

Assuming that is continuous, we can use Theorem 10.2.5 to write the arc length as

Therefore the length of a curve with polar equation , , is

Find the length of the cardioid .

SOLUTION The cardioid is shown in Figure 8. (We sketched it in Example 7 inSection 10.3.) Its full length is given by the parameter interval , so Formula 5 gives

We could evaluate this integral by multiplying and dividing the integrand by, or we could use a computer algebra system. In any event, we find that the

length of the cardioid is .

� � dr

d�2

sin2� � 2rdr

d�sin � cos � � r 2 cos2�

� dr

d�2

� r 2

f

L � yb

a� dx

d�2

� dy

d�2

d�

cos2� � sin2� � 1

dx

d�2

� dy

d�2

� dr

d�2

cos2� � 2rdr

d�cos � sin � � r 2 sin2�

r � f �� a � � � b

5 L � yb

a�r 2 � dr

d�2

d�

r � 1 � sin �

0 � � � 2�

L � y2�

0�r 2 � dr

d�2

d� � y2�

0s�1 � sin ��2 � cos2� d�

� y2�

0s2 � 2 sin � d�

s2 � 2 sin �L � 8

v EXAMPLE 4

O

FIGURE 8r=1+sin ¨


1–4 Find the area of the region that is bounded by the given curveand lies in the specified sector.

1. ,

2. ,

3. , ,

4. ,

r 2 � 9 sin 2� 0 � � � ��2

r � tan � ��6 � � � ��3

r � 0

r � � 2 0 � � � ��4

r � e ��2 � � � � 2�

5–8 Find the area of the shaded region.

5. 6.

r=œ„̈ r=1+cos ¨

10.4 Exercises


SECTION 10.4 AREAS AND LENGTHS IN POLAR COORDINATES 693

7. 8.

9–12 Sketch the curve and find the area that it encloses.

9. 10.

11. 12.

; 13–16 Graph the curve and find the area that it encloses.

13. 14.

15. 16.

17–21 Find the area of the region enclosed by one loop of the curve.

17. 18.

19. 20.

21. (inner loop)

22. Find the area enclosed by the loop of the strophoid.

23–28 Find the area of the region that lies inside the first curveand outside the second curve.

23. , 24. ,

25. ,

26. ,

27. ,

28. ,

29–34 Find the area of the region that lies inside both curves.

29. ,

30. ,

31. ,

32. ,

33. ,

34. , , ,

r=4+3 sin ¨ r=sin 2̈

r � 1 � sin �r � 2 sin �

r � 4 � 3 sin �r � 3 � 2 cos �

r � 3 � 2 cos 4�r � 2 � sin 4�

r � s1 � cos2�5�� r � 1 � 5 sin 6�

r 2 � sin 2�r � 4 cos 3�

r � sin 4� r � 2 sin 5�

r � 1 � 2 sin �

r � 2 cos � � sec �

r � 1r � 1 � sin �r � 1r � 2 cos �

r � 2 � sin �

r � 2r 2 � 8 cos 2�

r � 3 sin �

r � 1 � cos �r � 3 cos �

r � 2 � sin �r � 3 sin �

r � sin �r � s3 cos �

r � 1 � cos �r � 1 � cos �

r � cos 2�r � sin 2�

r � 3 � 2 sin �r � 3 � 2 cos �

r 2 � sin 2� r 2 � cos 2�

r � a sin � r � b cos � a 0 b 0

35. Find the area inside the larger loop and outside the smallerloop of the limaçon .

36. Find the area between a large loop and the enclosed smallloop of the curve .

37–42 Find all points of intersection of the given curves.

37. ,

38. ,

39. ,

40. ,

41. ,

42. ,

; 43. The points of intersection of the cardioid andthe spiral loop , , can’t be foundexactly. Use a graphing device to find the approximate valuesof at which they intersect. Then use these values to esti-mate the area that lies inside both curves.

44. When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it sup-presses noise from the audience. Suppose the microphone isplaced 4 m from the front of the stage (as in the figure) andthe boundary of the optimal pickup region is given by thecardioid , where is measured in meters andthe microphone is at the pole. The musicians want to knowthe area they will have on stage within the optimal pickuprange of the microphone. Answer their question.

45–48 Find the exact length of the polar curve.

45. ,

46. ,

47. ,

48.

; 49–50 Find the exact length of the curve. Use a graph todetermine the parameter interval.

49. 50.

r � 1 � 2 cos 3�

r � 1 � sin � r � 3 sin �

r � 1 � cos � r � 1 � sin �

r � 2 sin 2� r � 1

r � cos 3� r � sin 3�

r � sin � r � sin 2�

r 2 � sin 2� r 2 � cos 2�

r � 1 � sin �r � 2� ��2 � � � ��2

�

r � 8 � 8 sin � r

stage

audiencemicrophone

12 m

4 m

r � 2 cos � 0 � � � �

r � 5� 0 � � � 2�

r � � 2 0 � � � 2�

r � 2�1 � cos ��

r � cos4��4� r � cos2��2�

r � 12 � cos �



51–54 Use a calculator to find the length of the curve correct tofour decimal places. If necessary, graph the curve to determine theparameter interval.

51. One loop of the curve

52. ,

53.

54.

55. (a) Use Formula 10.2.6 to show that the area of the surfacegenerated by rotating the polar curve

r � cos 2�

��6 � � � ��3r � tan �

r � sin��4�

r � sin�6 sin ��

a � � � br � f ��

(where is continuous and ) about thepolar axis is

(b) Use the formula in part (a) to find the surface areagenerated by rotating the lemniscate about the polar axis.

56. (a) Find a formula for the area of the surface generated byrotating the polar curve , (where iscontinuous and ), about the line .

(b) Find the surface area generated by rotating the lemniscateabout the line .

r 2 � cos 2�

f a � � � br � f ��0 � a � b � � � � ��2

� � ��2r 2 � cos 2�

S � yb

a2�r sin ��r 2 � dr

d�2

d�

0 � a � b � �f

In this section we give geometric definitions of parabolas, ellipses, and hyperbolas andderive their standard equations. They are called conic sections, or conics, because theyresult from intersecting a cone with a plane as shown in Figure 1.

ParabolasA parabola is the set of points in a plane that are equidistant from a fixed point (calledthe focus) and a fixed line (called the directrix). This definition is illustrated by Figure 2.Notice that the point halfway between the focus and the directrix lies on the parabola; it iscalled the vertex. The line through the focus perpendicular to the directrix is called the axisof the parabola.

In the 16th century Galileo showed that the path of a projectile that is shot into the air at an angle to the ground is a parabola. Since then, parabolic shapes have been used in designing automobile headlights, reflecting telescopes, and suspension bridges. (SeeProblem 16 on page 196 for the reflection property of parabolas that makes them so useful.)

We obtain a particularly simple equation for a parabola if we place its vertex at the origin and its directrix parallel to the -axis as in Figure 3. If the focus is the point

, then the directrix has the equation . If is any point on the parabola,

FIGURE 1Conics

ellipse hyperbolaparabola

F

xOP�x, y�y � �p�0, p�

10.5 Conic Sections

axis

Ffocus

parabola

vertex directrix

FIGURE 2



1–8 Find the vertex, focus, and directrix of the parabola and sketchits graph.

1. 2.

3. 4.

5. 6.

7. 8.

9–10 Find an equation of the parabola. Then find the focus anddirectrix.

9. 10.

11–16 Find the vertices and foci of the ellipse and sketch its graph.

11. 12.

13. 14.

15.

16.

17–18 Find an equation of the ellipse. Then find its foci.

17. 18.

19–24 Find the vertices, foci, and asymptotes of the hyperbola andsketch its graph.

19. 20.

21. 22.

x 2 � 6y 2y 2 � 5x

2x � �y 2 3x 2 � 8y � 0

�x � 2�2 � 8�y � 3� x � 1 � �y � 5�2

y 2 � 2y � 12x � 25 � 0 y � 12x � 2x 2 � 16

y

x

1

_2

y

x

1

20

9x 2 � 18x � 4y 2 � 27

x 2 � 3y2 � 2x � 12y � 10 � 0

y

x

1

10

y

x

1

2

y 2

25�

x 2

9� 1

x 2

36�

y 2

64� 1

x 2 � y 2 � 100 y 2 � 16x 2 � 16

x 2

9�

y 2

5� 1

x 2

64�

y 2

100� 1

4x 2 � y 2 � 16 4x 2 � 25y 2 � 25

23.

24.

25–30 Identify the type of conic section whose equation is givenand find the vertices and foci.

25. 26.

27. 28.

29. 30.

31–48 Find an equation for the conic that satisfies the given conditions.

31. Parabola, vertex , focus

32. Parabola, focus , directrix

33. Parabola, focus , directrix

34. Parabola, focus , vertex

35. Parabola, vertex , vertical axis,passing through

36. Parabola, horizontal axis, passing through , , and

37. Ellipse, foci , vertices

38. Ellipse, foci , vertices

39. Ellipse, foci , , vertices ,

40. Ellipse, foci , , vertex

41. Ellipse, center , vertex , focus

42. Ellipse, foci , passing through

43. Hyperbola, vertices , foci

44. Hyperbola, vertices , foci

45. Hyperbola, vertices , , foci ,

46. Hyperbola, vertices , , foci ,

47. Hyperbola, vertices , asymptotes

48. Hyperbola, foci , , asymptotes and

x 2 � y � 1 x 2 � y 2 � 1

x 2 � 4y � 2y 2 y 2 � 8y � 6x � 16

y 2 � 2y � 4x 2 � 3 4x 2 � 4x � y 2 � 0

�0, 0� �1, 0�

�0, 0� y � 6

��4, 0� x � 2

�3, 6� �3, 2�

�2, 3��1, 5�

��1, 0� �1, �1� �3, 1�

��2, 0� ��5, 0�

�0, �5� �0, �13�

�0, 2� �0, 6� �0, 0� �0, 8�

�0, �1� �8, �1� �9, �1�

��1, 4� ��1, 0� ��1, 6�

��4, 0� ��4, 1.8�

��3, 0� ��5, 0�

�0, �2� �0, �5�

��3, �4� ��3, 6��3, �7� ��3, 9�

��1, 2� �7, 2��2, 2� �8, 2�

��3, 0� y � �2x

�2, 0� �2, 8�y � 3 �

12 x y � 5 �

12 x

y2 � 4x 2 � 2y � 16x � 31

4x 2 � y2 � 24x � 4y � 28 � 0

10.5 Exercises

1. Homework Hints available at stewartcalculus.com


SECTION 10.5 CONIC SECTIONS 701

49. The point in a lunar orbit nearest the surface of the moon iscalled perilune and the point farthest from the surface is calledapolune. The Apollo 11 spacecraft was placed in an ellipticallunar orbit with perilune altitude 110 km and apolune altitude314 km (above the moon). Find an equation of this ellipse ifthe radius of the moon is 1728 km and the center of the moonis at one focus.

50. A cross-section of a parabolic reflector is shown in the figure.The bulb is located at the focus and the opening at the focus is 10 cm.(a) Find an equation of the parabola.(b) Find the diameter of the opening , 11 cm from

the vertex.

51. In the LORAN (LOng RAnge Navigation) radio navigationsystem, two radio stations located at and transmit simul ta-neous signals to a ship or an aircraft located at . The onboardcomputer converts the time difference in receiving these signalsinto a distance difference , and this, according tothe definition of a hyperbola, locates the ship or aircraft on onebranch of a hyperbola (see the figure). Suppose that station B islocated 400 mi due east of station A on a coastline. A shipreceived the signal from B 1200 micro seconds (�s) before itreceived the signal from A.(a) Assuming that radio signals travel at a speed of 980 ft �s,

find an equation of the hyperbola on which the ship lies.(b) If the ship is due north of , how far off the coastline is

the ship?

52. Use the definition of a hyperbola to derive Equation 6 for ahyperbola with foci and vertices .

53. Show that the function defined by the upper branch of thehyperbola is concave upward.

� CD �

5 cm

5 cm

A

B

C

D

VF

11 cm

BAP

� PA � � � PB �

�

B

400 mitransmitting stations

coastlineA B

P

��a, 0��c, 0�

y 2�a 2 � x 2�b 2 � 1

54. Find an equation for the ellipse with foci andand major axis of length 4.

55. Determine the type of curve represented by the equation

in each of the following cases: (a) , (b) , and (c) .(d) Show that all the curves in parts (a) and (b) have the same

foci, no matter what the value of is.

56. (a) Show that the equation of the tangent line to the parabola at the point can be written as

(b) What is the -intercept of this tangent line? Use this fact todraw the tangent line.

57. Show that the tangent lines to the parabola drawnfrom any point on the directrix are perpendicular.

58. Show that if an ellipse and a hyperbola have the same foci,then their tangent lines at each point of intersection are perpendicular.

59. Use parametric equations and Simpson’s Rule with toestimate the circumference of the ellipse .

60. The planet Pluto travels in an elliptical orbit around the sun (at one focus). The length of the major axis is kmand the length of the minor axis is km. Use Simp-son’s Rule with to estimate the distance traveled by theplanet during one complete orbit around the sun.

61. Find the area of the region enclosed by the hyperbolaand the vertical line through a focus.

62. (a) If an ellipse is rotated about its major axis, find the volumeof the resulting solid.

(b) If it is rotated about its minor axis, find the resultingvolume.

63. Find the centroid of the region enclosed by the -axis and thetop half of the ellipse .

64. (a) Calculate the surface area of the ellipsoid that is generatedby rotating an ellipse about its major axis.

(b) What is the surface area if the ellipse is rotated about itsminor axis?

65. Let be a point on the ellipse withfoci and and let and be the angles between the lines

x 2

k�

y 2

k � 16� 1

��1, �1��1, 1�

0 � k � 16k � 16k � 0

k

�x0, y0�y 2 � 4px

y0y � 2p�x � x 0�

x

x 2 � 4py

n � 89x 2 � 4y 2 � 36

1.18 � 1010

1.14 � 1010

n � 10

x 2�a 2 � y 2�b 2 � 1

x9x 2 � 4y 2 � 36

x 2�a 2 � y 2�b 2 � 1P�x1, y1�F2F1


In the preceding section we defined the parabola in terms of a focus and directrix, but wedefined the ellipse and hyperbola in terms of two foci. In this section we give a more uni-fied treatment of all three types of conic sections in terms of a focus and directrix. Further- more, if we place the focus at the origin, then a conic section has a simple polar equation,which provides a convenient description of the motion of planets, satellites, and comets.

Theorem Let be a fixed point (called the focus) and be a fixed line (calledthe directrix) in a plane. Let be a fixed positive number (called the eccentricity).The set of all points in the plane such that

(that is, the ratio of the distance from to the distance from is the constant ) is a conic section. The conic is

(a)

(b)

(c)

PROOF Notice that if the eccentricity is , then and so the given condi-tion simply becomes the definition of a parabola as given in Section 10.5.

F le

P

� PF �� Pl � � e

F l e

an ellipse if e � 1

a parabola if e � 1

1

a hyperbola if e � 1

� PF � � � Pl �e � 1


, and the ellipse as shown in the figure. Prove that. This explains how whispering galleries and litho tripsy

work. Sound coming from one focus is reflected and passesthrough the other focus. [Hint: Use the formula in Problem 15on page 195 to show that .]

66. Let be a point on the hyperbolawith foci and and let and be the angles between the lines , and the hyperbola as shown in the figure.Prove that . (This is the reflection property of the

PF2PF1

�

tan � tan

F¡ F™0 x

y

∫

å

+ =1≈

a@

¥

b@

P(⁄, ›)

x 2�a 2 � y 2�b 2 � 1P�x1, y1�F2F1

PF2PF1

�

hyperbola. It shows that light aimed at a focus of a hyper-bolic mirror is reflected toward the other focus .)

0 x

y

å∫

F™F¡

P

F™F¡

P

F2

F1

10.6 Conic Sections in Polar Coordinates



1–8 Write a polar equation of a conic with the focus at the originand the given data.

1. Ellipse, eccentricity , directrix

2. Parabola, directrix

3. Hyperbola, eccentricity 1.5, directrix

4. Hyperbola, eccentricity 3, directrix

5. Parabola, vertex

6. Ellipse, eccentricity , vertex

7. Ellipse, eccentricity , directrix

8. Hyperbola, eccentricity 3, directrix

9–16 (a) Find the eccentricity, (b) identify the conic, (c) give anequation of the directrix, and (d) sketch the conic.

9. 10.

11. 12.

13. 14.

15. 16.

; 17. (a) Find the eccentricity and directrix of the conicand graph the conic and its directrix.

(b) If this conic is rotated counterclockwise about the originthrough an angle , write the resulting equation andgraph its curve.

; 18. Graph the conic and its directrix. Alsograph the conic obtained by rotating this curve about the ori-gin through an angle .

; 19. Graph the conics with , , , and on a common screen. How does the value of

affect the shape of the curve?

; 20. (a) Graph the conics for and var-ious values of . How does the value of affect the shapeof the conic?

(b) Graph these conics for and various values of .How does the value of affect the shape of the conic?

21. Show that a conic with focus at the origin, eccentricity , anddirectrix has polar equation

x � �3

y � 2

x � 3

�4, 3��2�

0.8 �1, ��2�12 r � 4 sec �

r � �6 csc �

12 x � 4

r �4

5 � 4 sin �r �

12

3 � 10 cos �

r �3

2 � 2 cos �

r �9

6 � 2 cos �r �

8

4 � 5 sin �

r �3

4 � 8 cos �r �

10

5 � 6 sin �

r � 1��1 � 2 sin ��

3��4

r � 4��5 � 6 cos ��

��3

r � e��1 � e cos � � e � 0.4 0.60.8 1.0 e

r � ed��1 � e sin �� e � 1d d

d � 1 ee

ex � �d

r �ed

1 � e cos �

r �1

1 � sin �



24. Show that the parabolas andintersect at right angles.

25. The orbit of Mars around the sun is an ellipse with eccen-tricity and semimajor axis . Find a polarequation for the orbit.

26. Jupiter’s orbit has eccentricity and the length of themajor axis is . Find a polar equation for theorbit.

27. The orbit of Halley’s comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and onefocus at the sun. The length of its major axis is 36.18 AU. [An astronomical unit (AU) is the mean distance between theearth and the sun, about 93 million miles.] Find a polar equa-tion for the orbit of Halley’s comet. What is the maximumdistance from the comet to the sun?

28. The Hale-Bopp comet, discovered in 1995, has an ellipticalorbit with eccentricity 0.9951 and the length of the majoraxis is 356.5 AU. Find a polar equation for the orbit of thiscomet. How close to the sun does it come?

29. The planet Mercury travels in an elliptical orbit with eccen-tricity . Its minimum distance from the sun is

km. Find its maximum distance from the sun.

30. The distance from the planet Pluto to the sun is km at perihelion and km at aphelion.

Find the eccentricity of Pluto’s orbit.

31. Using the data from Exercise 29, find the distance traveled bythe planet Mercury during one complete orbit around the sun.(If your calculator or computer algebra system evaluates defi-nite integrals, use it. Otherwise, use Simpson’s Rule.)

ey � d

r �ed

1 � e sin �

ey � �d

r �ed

1 � e sin �

r � c��1 � cos ��r � d��1 � cos ��

0.093 2.28 � 108 km

0.0481.56 � 109 km

0.2064.6 � 107

4.43 � 109 7.37 � 109

10.6 Exercises


© D

ean

Kete

lsen


CHAPTER 10 REVIEW 709

10 Review

1. (a) What is a parametric curve?(b) How do you sketch a parametric curve?

2. (a) How do you find the slope of a tangent to a parametriccurve?

(b) How do you find the area under a parametric curve?

3. Write an expression for each of the following:(a) The length of a parametric curve(b) The area of the surface obtained by rotating a parametric

curve about the

4. (a) Use a diagram to explain the meaning of the polar coordi-nates of a point.

(b) Write equations that express the Cartesian coordinates of a point in terms of the polar coordinates.

(c) What equations would you use to find the polar coordi natesof a point if you knew the Cartesian coordinates?

5. (a) How do you find the slope of a tangent line to a polarcurve?

(b) How do you find the area of a region bounded by a polarcurve?

(c) How do you find the length of a polar curve?

x-axis

�r, ��

�x, y�

6. (a) Give a geometric definition of a parabola.(b) Write an equation of a parabola with focus and direc-

trix . What if the focus is and the directrix is ?

7. (a) Give a definition of an ellipse in terms of foci.(b) Write an equation for the ellipse with foci and

vertices .

8. (a) Give a definition of a hyperbola in terms of foci.(b) Write an equation for the hyperbola with foci and

vertices .(c) Write equations for the asymptotes of the hyperbola in

part (b).

9. (a) What is the eccentricity of a conic section?(b) What can you say about the eccentricity if the conic section

is an ellipse? A hyperbola? A parabola?(c) Write a polar equation for a conic section with eccentricity

and directrix . What if the directrix is ?? ?

�0, p�y � �p �p, 0�

x � �p

��c, 0��a, 0�

��c, 0��a, 0�

e x � d x � �dy � d y � �d

Concept Check

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

1. If the parametric curve , satisfies ,then it has a horizontal tangent when .

2. If and are twice differentiable, then

3. The length of the curve , , , is.

4. If a point is represented by in Cartesian coordinates(where ) and in polar coordinates, then

.

x � f �t� y � t�t� t��1� � 0t � 1

y � t�t�x � f �t�

d 2y

dx 2 �d 2y�dt 2

d 2x�dt 2

a � t � by � t�t�x � f �t�x

ba s� f ��t� 2 � �t��t� 2 dt

�x, y��r, ��x � 0

� � tan �1� y�x�

5. The polar curves and have thesame graph.

6. The equations , , and ,all have the same graph.

7. The parametric equations , have the same graphas , .

8. The graph of is a parabola.

9. A tangent line to a parabola intersects the parabola only once.

10. A hyperbola never intersects its directrix.

r � 2 x 2 � y 2 � 4 x � 2 sin 3ty � 2 cos 3t �0 � t � 2��

x � t 2 y � t 4

x � t 3 y � t 6

y 2 � 2y � 3x

r � sin 2� � 1r � 1 � sin 2�

True-False Quiz



; Graphing calculator or computer required Computer algebra system requiredCAS

1–4 Sketch the parametric curve and eliminate the parameter tofind the Cartesian equation of the curve.

1. , ,

2. ,

3. , ,

4. ,

5. Write three different sets of parametric equations for the curve .

6. Use the graphs of and to sketch the para-metric curve , . Indicate with arrows the direction in which the curve is traced as increases.

7. (a) Plot the point with polar coordinates . Then findits Cartesian coordinates.

(b) The Cartesian coordinates of a point are . Find twosets of polar coordinates for the point.

8. Sketch the region consisting of points whose polar coor-dinates satisfy .

9–16 Sketch the polar curve.

9. 10.

11. 12.

13. 14.

15. 16.

17–18 Find a polar equation for the curve represented by thegiven Cartesian equation.

17. 18.

; 19. The curve with polar equation is called acochleoid. Use a graph of as a function of in Cartesiancoordinates to sketch the cochleoid by hand. Then graph itwith a machine to check your sketch.

; 20. Graph the ellipse and its directrix. Also graph the ellipse obtained by rotation about the originthrough an angle .

�4 � t � 1y � 2 � tx � t 2 � 4t

y � e tx � 1 � e 2 t

0 � � � ��2y � sec �x � cos �

y � 1 � sin �x � 2 cos �

y � sx

y � t�t�x � f �t�y � t�t�x � f �t�

t

t

x

_1

1 t

y

1

1

�4, 2��3�

��3, 3�

1 � r � 2 and ��6 � � � 5��6

r � sin 4�r � 1 � cos �

r � 3 � cos 3�r � cos 3 �

r � 2 cos��2�r � 1 � cos 2�

r �3

2 � 2 cos �r �

3

1 � 2 sin �

x 2 � y 2 � 2x � y � 2

r � �sin � ��r

r � 2��4 � 3 cos � �

2��3

21–24 Find the slope of the tangent line to the given curve at thepoint corresponding to the specified value of the parameter.

21. , ;

22. , ;

23. ;

24. ;

25–26 Find and .

25. ,

26. ,

; 27. Use a graph to estimate the coordinates of the lowest point onthe curve , . Then use calculus tofind the exact coordinates.

28. Find the area enclosed by the loop of the curve in Exercise 27.

29. At what points does the curve

have vertical or horizontal tangents? Use this information tohelp sketch the curve.

30. Find the area enclosed by the curve in Exercise 29.

31. Find the area enclosed by the curve .

32. Find the area enclosed by the inner loop of the curve.

33. Find the points of intersection of the curves and.

34. Find the points of intersection of the curves and.

35. Find the area of the region that lies inside both of the circlesand .

36. Find the area of the region that lies inside the curvebut outside the curve .

37–40 Find the length of the curve.

37. , ,

38. , ,

39. ,

40. ,

x � ln t y � 1 � t 2 t � 1

x � t 3 � 6t � 1 y � 2t � t 2 t � �1

r � e ��

r � 3 � cos 3� � � ��2

dy�dx d 2 y�dx 2

x � t � sin t y � t � cos t

x � 1 � t 2 y � t � t 3

x � t 3 � 3t y � t 2 � t � 1

x � 2a cos t � a cos 2t y � 2a sin t � a sin 2t

r 2 � 9 cos 5�

r � 1 � 3 sin �

r � 2r � 4 cos �

r � cot �r � 2 cos �

r � 2 sin � r � sin � � cos �

r � 2 � cos 2� r � 2 � sin �

x � 3t 2 y � 2t 3 0 � t � 2

x � 2 � 3t y � cosh 3t 0 � t � 1

r � 1�� 2�

r � sin3��3� 0 � � � �

Exercises


CHAPTER 10 REVIEW 711

41–42 Find the area of the surface obtained by rotating the givencurve about the -axis.

41. , ,

42. , ,

; 43. The curves defined by the parametric equations

are called strophoids (from a Greek word meaning “to turnor twist”). Investigate how these curves vary as varies.

; 44. A family of curves has polar equations where is a positive number. Investigate how the curves change as changes.

45–48 Find the foci and vertices and sketch the graph.

45. 46.

47.

48.

49. Find an equation of the ellipse with foci and vertices.

50. Find an equation of the parabola with focus and direc-trix .

51. Find an equation of the hyperbola with foci andasymptotes .

52. Find an equation of the ellipse with foci and majoraxis with length 8.

x

x � 4st y �t 3

3�

1

2t 2 1 � t � 4

x � 2 � 3t y � cosh 3t 0 � t � 1

x �t 2 � c

t 2 � 1y �

t�t 2 � c�t 2 � 1

c

r a � � sin 2� �aa

x 2

9�

y 2

8� 1 4x 2 � y 2 � 16

6y 2 � x � 36y � 55 � 0

25x 2 � 4y 2 � 50x � 16y � 59

��4, 0��5, 0�

�2, 1�x � �4

�0, �4�y � �3x

�3, �2�

53. Find an equation for the ellipse that shares a vertex and afocus with the parabola and that has its otherfocus at the origin.

54. Show that if is any real number, then there are exactly two lines of slope that are tangent to the ellipse

and their equations are.

55. Find a polar equation for the ellipse with focus at the origin,eccentricity .

56. Show that the angles between the polar axis and the asymptotes of the hyperbola , , are given by .

57. A curve called the folium of Descartes is defined by theparametric equations

(a) Show that if lies on the curve, then so does ;that is, the curve is symmetric with respect to the line

. Where does the curve intersect this line?(b) Find the points on the curve where the tangent lines are

horizontal or vertical.(c) Show that the line is a slant asymptote.(d) Sketch the curve.(e) Show that a Cartesian equation of this curve is

.(f ) Show that the polar equation can be written in the form

(g) Find the area enclosed by the loop of this curve.(h) Show that the area of the loop is the same as the area that

lies between the asymptote and the infinite branches ofthe curve. (Use a computer algebra system to evaluate the integral.)

y � x

y � �x � 1

x 3 � y 3 � 3xy

r �3 sec � tan �

1 � tan3�

CAS

x 2 � y � 100

mm

x 2�a 2 � y 2�b 2 � 1y � mx � sa 2m 2 � b 2

13 , and directrix with equation r � 4 sec �

e 1r � ed��1 � e cos ��cos�1��1�e�

x �3t

1 � t 3 y �3t 2

1 � t 3

�b, a��a, b�


Problems Plus1. A curve is defined by the parametric equations

Find the length of the arc of the curve from the origin to the nearest point where there is a verti-cal tangent line.

2. (a) Find the highest and lowest points on the curve .(b) Sketch the curve. (Notice that it is symmetric with respect to both axes and both of the lines

, so it suffices to consider initially.)(c) Use polar coordinates and a computer algebra system to find the area enclosed by the curve.

; 3. What is the smallest viewing rectangle that contains every member of the family of polar curves, where ? Illustrate your answer by graphing several members of the

family in this viewing rectangle.

4. Four bugs are placed at the four corners of a square with side length . The bugs crawl counter-clockwise at the same speed and each bug crawls directly toward the next bug at all times. Theyapproach the center of the square along spiral paths.(a) Find the polar equation of a bug’s path assuming the pole is at the center of the square. (Use

the fact that the line joining one bug to the next is tangent to the bug’s path.)(b) Find the distance traveled by a bug by the time it meets the other bugs at the center.

5. Show that any tangent line to a hyperbola touches the hyperbola halfway between the points ofintersection of the tangent and the asymptotes.

6. A circle of radius has its center at the origin. A circle of radius rolls without slipping inthe counterclockwise direction around . A point is located on a fixed radius of the rollingcircle at a distance from its center, . [See parts (i) and (ii) of the figure.] Let bethe line from the center of to the center of the rolling circle and let be the angle thatmakes with the positive -axis.(a) Using as a parameter, show that parametric equations of the path traced out by are

Note: If , the path is a circle of radius ; if , the path is an epicycloid. The pathtraced out by for is called an epitrochoid.

; (b) Graph the curve for various values of between and .

(c) Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid ison the circle of radius centered at the origin.

Note: This is the principle of the Wankel rotary engine. When the equilateral triangle rotateswith its vertices on the epitrochoid, its centroid sweeps out a circle whose center is at thecenter of the curve.

(d) In most rotary engines the sides of the equilateral triangles are replaced by arcs of circlescentered at the opposite vertices as in part (iii) of the figure. (Then the diameter of the rotoris constant.) Show that the rotor will fit in the epitrochoid if .

x � yt

1

cos u

udu y � y

t

1

sin u

udu

x 4 � y 4 � x 2 � y 2

y x 0y � �xCAS

0 � c � 1r � 1 � c sin �

a

r2rCPC

L0 � b � rbL�C

xP�

x � b cos 3� � 3r cos � y � b sin 3� � 3r sin �

b � r3rb � 00 � b � rP

r0b

b

b �32 (2 � s3 )r

(ii)

y

xP¸¨

P

y

x

r

b

P=P¸

2r

(i) (iii)

712

a

a a

a

FIGURE FOR PROBLEM 4


Documents

SECTION 10.1 665 - NCKU